Symplectic automorphism conjecture for the Albanese kernel of zero-cycles

Let $S$ be a smooth projective surface. A symplectic automorphism is an automorphism acting trivially on the space of holomorphic two-forms $H^{2,0}(S)$. It acts on the Albanese kernel $\mathrm{CH}_0(S)_{\mathrm{alb}}$ of the degree-zero zero-cycle group. Symplectic automorphism conjecture. Any symplectic automorphism acts trivially on $\mathrm{CH}_0(S)_{\mathrm{alb}}$. This is presented as a special, more tractable case of the preceding correspondence conjecture. The paper's abstract indicates that the claim is proved for the relevant elliptic surfaces except for specified exceptional cases, but the general surface statement is not established in the supplied text.